In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.

Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces

In this paper, we study strong convergence of some proximal-type algorithms to a solution of split minimization problem in complete p-uniformly convex metric spaces. We also analyse asymptotic behaviour of the sequence generated by Halpern-type proximal point algorithm and extend it to approximate a common solution of a finite family of minimization problems in the setting of complete p-uniformly convex metric spaces. Furthermore, numerical experiments of our algorithms in comparison with other algorithms are given to show the applicability of our results.

/pdf/proximal-type-algorithms-for-split-minimization-problem-in-p-qsl9lldzwq.pdf

Proximal-type algorithms for split minimization problem in P-uniformly convex metric spaces

In this paper, we propose a new inertial extrapolation method for solving the generalized split feasibility problems over the solution set of monotone variational inclusion problems in real Hilbert...

An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions

In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction.

An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings

/pdf/inertial-shrinking-projection-algorithm-with-self-adaptive-1i5cmgj3yj.pdf

Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings

In this paper, we introduce a new algorithm with self adaptive step-size for finding a common solution of a split feasibility problem and a fixed point problem in real Hilbert spaces. Motivated by the self adaptive step-size method, we incorporate the self adaptive step-size to overcome the difficulty of having to compute the operator norm in the proposed method. Under standard and mild assumption on the control sequences, we establish the strong convergence of the algorithm, obtain a common element in the solution set of a split feasibility problem for sum of two monotone operators and fixed point problem of a demimetric mapping. Numerical examples are presented to illustrate the performance and the behavior of our method. Our result extends, improves and unifies other results in the literature.

A strong convergence algorithm for a fixed point constrained split null point problem

Abstract In this paper, we propose a viscosity-type algorithm to approximate a common solution of a monotone inclusion problem, a minimization problem and a fixed point problem for an infinitely countable family of ( f , g ) {(f,g)} -generalized k-strictly pseudononspreading mappings in a CAT ⁢ ( 0 ) {\\mathrm{CAT}(0)} space. We obtain a strong convergence of the proposed algorithm to the aforementioned problems in a complete CAT ⁢ ( 0 ) {\\mathrm{CAT}(0)} space. Furthermore, we give an application of our result to a nonlinear Volterra integral equation and a numerical example to support our main result. Our results complement and extend many recent results in literature.

A viscosity-type algorithm for an infinitely countable family of (f,g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces

The main purpose of this paper is to introduce some viscosity-type proximal point algorithms which comprise of a nonexpansive mapping and a finite sum of resolvents of monotone operators, and prove their strong convergence to a common zero of a finite family of monotone operators which is also a fixed point of a nonexpansive mapping and a unique solution of some variational inequality problems in an Hadamard space. We apply our results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems.

Viscosity iterative techniques for approximating a common zero of monotone operators in an Hadamard space

The main purpose of this paper is to introduce a modified inertial forward-backward splitting method and prove its strong convergence to a zero of the sum of two accretive operators in real uniformly convex Banach space which is also uniformly smooth. We then apply our results to solve variational inequality problem and convex minimization problem. We also give a numerical example of our algorithm to show that it converges faster than the un-accelerated modified forwardbackward algorithm.

/pdf/strong-convergence-of-an-inertial-forward-backward-splitting-2p6ujelgwy.pdf

Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space

In this paper, we propose a generalized viscosity iterative algorithm which includes a sequence of contractions and a self adaptive step size for approximating a common solution of a multiple-set split feasibility problem and fixed point problem for countable families of k-strictly pseudononspeading mappings in the framework of real Hilbert spaces. The advantage of the step size introduced in our algorithm is that it does not require the computation of the Lipschitz constant of the gradient operator which is very difficult in practice. We also introduce an inertial process version of the generalize viscosity approximation method with self adaptive step size. We prove strong convergence results for the sequences generated by the algorithms for solving the aforementioned problems and present some numerical examples to show the efficiency and accuracy of our algorithm. The results presented in this paper extends and complements many recent results in the literature.

https://www.mdpi.com/2075-1680/10/1/1/pdf

An inertial generalized viscosity approximation method for solving multiple-sets split feasibility problems and common fixed point of strictly pseudo-nonspreading mappings

H. A. Abass

Papers

A strong convergence algorithm for a fixed point constrained split null point problem

A viscosity-type algorithm for an infinitely countable family of (f,g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces

Viscosity iterative techniques for approximating a common zero of monotone operators in an Hadamard space

Strong convergence of an inertial forward-backward splitting method for accretive operators in real Banach space

An inertial generalized viscosity approximation method for solving multiple-sets split feasibility problems and common fixed point of strictly pseudo-nonspreading mappings